# Fast Fourier Transform

Chapter

First Online:

- 916 Downloads

## Abstract

If Equation (2) is called the

*f*is an integrable function of a variable \(t\in (-\ ,\,\infty )\), then its**Fourier integral**or**Fourier transform**may be defined as the complex function*F*of a variable \(\omega \in (-\infty ,\,\infty )\) where \(F(\omega ):=\int _{-\infty }^{\infty }f(t)\,e^{-i\omega t}\, dt,\quad i=\sqrt{-1}\). The most important fact about definition (1) is that if it is viewed as an integral equation, its solution under very general conditions is$$f(t)=\frac{1}{2\pi }\int _{-\infty }^{\infty }F(\omega )\,e^{i\omega t}\, d\omega (2)$$

**inverse transform**of the**direct transform**of (1), and the two constitute the Fourier transform pair.## Copyright information

© Springer Nature Singapore Pte Ltd. 2019